Actually they do. The zodiac sign constellation of a day correspond to the constellation over which the sun while be at noon. So you do have a zodiac sign constellation associated to your day of bird. But depending on how your culture group stars in constellations you might be from a different zodiac. If you want to follow a classical westerner map of the sky, they will be 13 of them.
Also, note that the zodiac signs won’t cut the year in equal parts. Some constellations are just bigger than other and the sun path across the sky changes more or less quickly from a day to another depending on the time of the year.